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          CPP优雅的生成全排列
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        <p><strong>为何会有这个问题？</strong></p>
<p>网易笔试很悲催的遇到这个问题，然后脑子生锈用很蠢的循环去写，而且还没写出来，这里我整理了一些其他的办法来优化，争取做到 高效，简洁，优美。</p>
<a id="more"></a>
<h3 id="方法一-stl的next_permutation">方法一 ：STL的next_permutation</h3>
<p>使用简洁，优美，但是不够高效，生成全部排列的复杂度大概为 $ O(n!*n) $,每次生成的复杂度为O(n). <a href="https://zh.cppreference.com/w/cpp/algorithm/next_permutation" target="_blank" rel="noopener">next_permutation</a></p>
<p>注意，生成前需要排序，按照$ &lt; <span class="math inline">\(的顺序排序。或者自定义comp对象。\)</span> (fisrt, last, comp) $</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;iostream&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;algorithm&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;vector&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">cout_nums</span><span class="params">(<span class="keyword">const</span> <span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; &amp;nums)</span></span>&#123; <span class="comment">//打印数组</span></span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;nums.size();i++)</span><br><span class="line">        <span class="built_in">cout</span> &lt;&lt;nums[i]&lt;&lt;<span class="string">" "</span>;</span><br><span class="line">    <span class="built_in">cout</span> &lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>&#123;</span><br><span class="line">    <span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; nums; <span class="comment">//数组</span></span><br><span class="line">    <span class="built_in">vector</span>&lt;<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;&gt; permutations;<span class="comment">//存储所有排列</span></span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;<span class="number">5</span>;i++) <span class="comment">//初始化数组</span></span><br><span class="line">        nums.push_back(i);</span><br><span class="line"></span><br><span class="line">    sort(nums.begin(),nums.end()); <span class="comment">//按照从小到大的顺序排序</span></span><br><span class="line">    permutations.push_back(<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;(nums.begin(),nums.end())); </span><br><span class="line">    <span class="keyword">while</span>(next_permutation(nums.begin(),nums.end()))<span class="comment">//生成下一个排列</span></span><br><span class="line">        permutations.push_back(<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;(nums.begin(), nums.end()));</span><br><span class="line"></span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;permutations.size();i++) <span class="comment">//打印所有排列</span></span><br><span class="line">        cout_nums(permutations[i]);</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="方法二-dfs-递归生成">方法二 DFS 递归生成</h3>
<p>速度理论最优<span class="math inline">\(O(n!)\)</span>,但实际和方法一速度差不多，都很慢。使用更简洁，不过实现时需要注意trick。</p>
<p>DFS就是模仿挑选的过程，值得注意的是这里采用<strong>交换来实现的挑选</strong>。</p>
<p>而为了保证生成不会重复，需要在<strong>额外进行一次交换来进行复位</strong>。</p>
<p>证明如下： <strong>交换， 递归调用，再交换序列会恢复原状。</strong></p>
<p>对于长度为1的情况: 只有一次循环，且只和本身交换一次，显然成立</p>
<p>​ c == 第一次交换 ==&gt; c =&gt; 递归调用(生成结果) == 第二次交换 ==&gt; c 显然符合</p>
<p>假设长度为n是会恢复原状,则长度n+1时</p>
<p>​ <span class="math inline">\(x, (n_1,..,n_n)\)</span> == 第一次交换(与元素<span class="math inline">\(n_k\)</span>交换) ==&gt; <span class="math inline">\(n_x， (n_1,n_2,...,n_{k-1},x,n_{k+1},...,n_n)\)</span></p>
<p>==&gt; 递归调用，长度为n的情况，结束后序列显然不变 == 第二次交换(将元素<span class="math inline">\(n_k\)</span> 放回) ==&gt; 得到原序列 <span class="math inline">\(x, (n_1,..,n_n)\)</span></p>
<p><strong>数列每次打乱都会恢复原状，因此每一次挑选都不会重复</strong>，不过结果是无序的</p>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br></pre></td><td class="code"><pre><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">perm</span><span class="params">(<span class="keyword">int</span> depth, <span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; &amp;nums, <span class="built_in">vector</span>&lt;<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;&gt; &amp;perms)</span></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (depth &gt;= nums.size())&#123;</span><br><span class="line">        perms.push_back(<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;(nums.begin(), nums.end()));</span><br><span class="line">        <span class="keyword">return</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">for</span> (<span class="keyword">int</span> i = depth; i &lt; nums.size(); i++)&#123;</span><br><span class="line">    	<span class="comment">//注意这里是两次交换，第一次交换挑选，第二次交换复位</span></span><br><span class="line">        swap(nums[depth], nums[i]);</span><br><span class="line">        perm(depth + <span class="number">1</span>, nums, perms);</span><br><span class="line">        swap(nums[depth], nums[i]);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">perm</span><span class="params">(<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; &amp;nums, <span class="built_in">vector</span>&lt;<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;&gt; &amp;perms)</span></span>&#123;</span><br><span class="line">    <span class="comment">//对nums内的数字生成全排列</span></span><br><span class="line">    perm(<span class="number">0</span>, nums, perms);</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h4 id="使用方法如下">使用方法如下：</h4>
<figure class="highlight cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br><span class="line">42</span><br><span class="line">43</span><br><span class="line">44</span><br><span class="line">45</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;iostream&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;algorithm&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">include</span> <span class="meta-string">&lt;vector&gt;</span></span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">cout_nums</span><span class="params">(<span class="keyword">const</span> <span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; &amp;nums)</span></span>&#123;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;nums.size();i++)</span><br><span class="line">        <span class="built_in">cout</span> &lt;&lt;nums[i]&lt;&lt;<span class="string">" "</span>;</span><br><span class="line">    <span class="built_in">cout</span> &lt;&lt;<span class="built_in">endl</span>;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">perm</span><span class="params">(<span class="keyword">int</span> depth, <span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; &amp;nums, <span class="built_in">vector</span>&lt;<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;&gt; &amp;perms)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span> (depth &gt;= nums.size())</span><br><span class="line">    &#123;</span><br><span class="line">        perms.push_back(<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;(nums.begin(), nums.end()));</span><br><span class="line">        <span class="keyword">return</span>;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">for</span> (<span class="keyword">int</span> i = depth; i &lt; nums.size(); i++)</span><br><span class="line">    &#123;</span><br><span class="line">        swap(nums[depth], nums[i]);</span><br><span class="line">        perm(depth + <span class="number">1</span>, nums, perms);</span><br><span class="line">        swap(nums[depth], nums[i]);</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">perm</span><span class="params">(<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; &amp;nums, <span class="built_in">vector</span>&lt;<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;&gt; &amp;perms)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    perm(<span class="number">0</span>, nums, perms);</span><br><span class="line">&#125;</span><br><span class="line"></span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span>&#123;</span><br><span class="line"></span><br><span class="line">    <span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt; nums;</span><br><span class="line">    <span class="built_in">vector</span>&lt;<span class="built_in">vector</span>&lt;<span class="keyword">int</span>&gt;&gt; permutations;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;<span class="number">4</span>;i++)</span><br><span class="line">        nums.push_back(i);</span><br><span class="line"></span><br><span class="line">    perm(nums,permutations);</span><br><span class="line"></span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">0</span>;i&lt;permutations.size();i++)</span><br><span class="line">        cout_nums(permutations[i]);</span><br><span class="line"></span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure>
<h3 id="结语">结语</h3>
<p>内置next_permutation或着采用 递归(DFS)+两次交换 实现。</p>

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